Tapered waveguides are useful for efficiently interconnecting waveguides that support modes of different dimensions. Semiconductor laser chips typically contain waveguides that tightly guide the optical mode, while the optical fiber that transmits the laser light guides weakly so that the modes match poorly if the laser is connected directly to the fiber. Many approaches have been used to improve the efficiency of such interconnections. Vertical, lateral, and refractive index tapers were demonstrated by * R. Logan, U.S. Pat. No. 3,978,426, Aug. 31, 1976, and * R. K. Winn et al., IEEE Trans. Microwave Theory and Techniques MTT23 92 (1975), and * P. G. Suchoski Jr., et al., J. Light. Technology, LT5 1246 (1987). Both waveguide claddings and fibers have been tapered; see * J. Hammer, U.S. Pat. No. 4773720, Sep. 27, 1988, and * H. Schneider, U.S. Pat. No. 4,795,228, Jan 3, 1989. Multiple stage tapers in the vertical and longitudinal directions were shown by * T. Koch, U.S. Pat. No. 4,932,032, Jun. 5, 1990 and * P. Melman, U.S. Pat. No. 5,261,017, Nov. 9, 1993. Pairs of coupled waveguides have been shown with single and multiple horizontal and vertical tapers by * Y. Shani et al., Appl. Phys. Lett. 55 2389 (1989), * E. Kapon, U.S. Pat. No. 5,078,516, Jan 7, 1992, * Zengerle et al., Elect. Lett. 28 631 (1992), * B. Stegmueller, U.S. Pat. No. 5,199,092, Mar. 30, 1993 and * R. Smith et al., IEEE Photon. Technology Lett. 8 1052 (1996). Segmented tapers are shown by Z. Weissman et al., J. Light. Technology, 11 1831 (1993), and R. Adar, U.S. Pat. No. 5,577,141, Nov. 19, 1996. A good deal of effort has also been expended recently on optimizing integrated tapers in semiconductor lasers. However, the practical constraints that apply to any specific implementation tend to require different coupling approaches depending on both materials and geometry: the processes used to manufacture and package a product must be compatible with each other, with the materials used, and with the component layout. A unique geometry of taper and waveguides is required to optimize the coupling between a laser chip that is to be butt coupled in a hybrid configuration to a planar waveguide chip.